37 research outputs found
Existence of common and upper frequently hypercyclic subspaces
We provide criteria for the existence of upper frequently hypercyclic
subspaces and for common hypercyclic subspaces, which include the following
consequences. There exist frequently hypercyclic operators with
upper-frequently hypercyclic subspaces and no frequently hypercyclic subspace.
On the space of entire functions, each differentiation operator induced by a
non-constant polynomial supports an upper frequently hypercyclic subspace, and
the family of its non-zero scalar multiples has a common hypercyclic subspace.
A question of Costakis and Sambarino on the existence of a common hypercyclic
subspace for a certain uncountable family of weighted shift operators is also
answered.Comment: 30 page
Two remarks on the set of recurrent vectors
We solve in the negative two open problems, related to the linear and
topological structure of the set of recurrent vectors, asked by Sophie Grivaux,
Alfred Peris and the first author of this paper. Firstly, we show that there
exist recurrent operators whose set of recurrent vectors is not dense lineable;
and secondly, we construct operators which are reiteratively recurrent and
cyclic, but whose set of reiteratively recurrent vectors is meager.Comment: 17 page